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Object detection, tracking and recognition in images are key problems in computer vision. This book provides the reader with a balanced treatment between the theory and practice of selected methods in these areas to make the book accessible to a range of researchers, engineers, developers and postgraduate students working in computer vision and related fields.
Key features:

Explains the main theoretical ideas behind each method (which are augmented with a rigorous mathematical derivation of the formulas), their implementation (in C++) and demonstrated working in real applications.
Places an emphasis on tensor and statistical based approaches within object detection and recognition.
Provides an overview of image clustering and classification methods which includes subspace and kernel based processing, mean shift and Kalman filter, neural networks, and k-means methods.
Contains numerous case study examples of mainly automotive applications.
Includes a companion website hosting full C++ implementation, of topics presented in the book as a software library, and an accompanying manual to the software platform.



Table of Contents

Preface xiii

Acknowledgements xv

Notations and Abbreviations xvii

1 Introduction 1

1.1 A Sample of Computer Vision 3

1.2 Overview of Book Contents 6

References 8

2 Tensor Methods in Computer Vision 9

2.1 Abstract 9

2.2 Tensor - A Mathematical Object 10

2.2.1 Main Properties of Linear Spaces 10

2.2.2 Concept of a Tensor 11

2.3 Tensor - A Data Object 13

2.4 Basic Properties of Tensors 15

2.4.1 Notation of Tensor Indices and Components 16

2.4.2 Tensor Products 18

2.5 Tensor Distance Measures 20

2.5.1 Overview of Tensor Distances 22

2.5.1.1 Computation of Matrix Exponent and Logarithm Functions 24

2.5.2 Euclidean Image Distance and Standardizing Transform 29

2.6 Filtering of Tensor Fields 33

2.6.1 Order Statistic Filtering of Tensor Data 33

2.6.2 Anisotropic Diffusion Filtering 36

2.6.3 IMPLEMENTATION of Diffusion Processes 40

2.7 Looking into Images with the Structural Tensor 44

2.7.1 Structural Tensor in Two-Dimensional Image Space 47

2.7.2 Spatio-Temporal Structural Tensor 50

2.7.3 Multichannel and Scale-Space Structural Tensor 52

2.7.4 Extended Structural Tensor 54

2.7.4.1 IMPLEMENTATION of the Linear and Nonlinear Structural Tensor 57

2.8 Object Representation with Tensor of Inertia and Moments 62

2.8.1 IMPLEMENTATION of Moments and their Invariants 65

2.9 Eigendecomposition and Representation of Tensors 68

2.10 Tensor Invariants 72

2.11 Geometry of Multiple Views: The Multifocal Tensor 72

2.12 Multilinear Tensor Methods 75

2.12.1 Basic Concepts of Multilinear Algebra 78

2.12.1.1 Tensor Flattening 78

2.12.1.2 IMPLEMENTATION Tensor Representation 84

2.12.1.3 The k-mode Product of a Tensor and a Matrix 95

2.12.1.4 Ranks of a Tensor 100

2.12.1.5 IMPLEMENTATION of Basic Operations on Tensors 101

2.12.2 Higher-Order Singular Value Decomposition (HOSVD) 112

2.12.3 Computation of the HOSVD 114

2.12.3.1 Implementation of the HOSVD Decomposition 119

2.12.4 HOSVD Induced Bases 121

2.12.5 Tensor Best Rank-1 Approximation 123

2.12.6 Rank-1 Decomposition of Tensors 126

2.12.7 Best Rank-(R1, R2, . . . , RP) Approximation 131

2.12.8 Computation of the Best Rank-(R1, R2, . . . , RP) Approximations 134

2.12.8.1 IMPLEMENTATION - Rank Tensor Decompositions 137

2.12.8.2 CASE STUDY - Data Dimensionality Reduction 145

2.12.9 Subspace Data Representation 149

2.12.10 Nonnegative Matrix Factorization 151

2.12.11 Computation of the Nonnegative Matrix Factorization 155

2.12.12 Image Representation with NMF 160

2.12.13 Implementation of the Nonnegative Matrix Factorization 162

2.12.14 Nonnegative Tensor Factorization 169

2.12.15 Multilinear Methods of Object Recognition 173

2.13 Closure 179

2.13.1 Chapter Summary 179

2.13.2 Further Reading 180

2.13.3 Problems and Exercises 181

References 182

3 Classification Methods and Algorithms 189

3.1 Abstract 189

3.2 Classification Framework 190

3.2.1 IMPLEMENTATION Computer Representation of Features 191

3.3 Subspace Methods for Object Recognition 194

3.3.1 Principal Component Analysis 195

3.3.1.1 Computation of the PCA 199

3.3.1.2 PCA for Multi-Channel Image Processing 210

3.3.1.3 PCA for Background Subtraction 214

3.3.2 Subspace Pattern Classification 215

3.4 Statistical Formulation of the Object Recognition 222

3.4.1 Parametric and Nonparametric Methods 222

3.4.2 Probabilistic Framework 222

3.4.3 Bayes Decision Rule 223

3.4.4 Maximum a posteriori Classification Scheme 224

3.4.5 Binary Classification Problem 226

3.5 Parametric Methods - Mixture of Gaussians 227

3.6 The Kalman Filter 233

3.7 Nonparametric Methods 236

3.7.1 Histogram Based Techniques 236

3.7.2 Comparing Histograms 239

3.7.3 IMPLEMENTATION - Multidimensional Histograms 243

3.7.4 Parzen Method 246

3.7.4.1 Kernel Based Methods 248

3.7.4.2 Nearest-Neighbor Method 250

3.8 The Mean Shift Method 251

3.8.1 Introduction to the Mean Shift 251

3.8.2 Continuously Adaptive Mean Shift Method (CamShift) 257

3.8.3 Algorithmic Aspects of the Mean Shift Tracking 259

3.8.3.1 Tracking of Multiple Features 259

3.8.3.2 Tracking of Multiple Objects 260

3.8.3.3 Fuzzy Approach to the CamShift 261

3.8.3.4 Discrimination with Background Information 262

3.8.3.5 Adaptive Update of the Classifiers 263

3.8.4 IMPLEMENTATION of the CamShift Method 264

3.9 Neural Networks 267

3.9.1 Probabilistic Neural Network 267

3.9.2 IMPLEMENTATION - Probabilistic Neural Network 270

3.9.3 Hamming Neural Network 274

3.9.4 IMPLEMENTATION of the Hamming Neural Network 278

3.9.5 Morphological Neural Network 282

3.9.5.1 IMPLEMENTATION of the Morphological Neural Network 285

3.10 Kernels in Vision Pattern Recognition 291

3.10.1 Kernel Functions 296

3.10.2 IMPLEMENTATION - Kernels 301

3.11 Data Clustering 306

3.11.1 The k-Means Algorithm 308

3.11.2 Fuzzy c-Means 311

3.11.3 Kernel Fuzzy c-Means 313

3.11.4 Measures of Cluster Quality 315

3.11.5 IMPLEMENTATION Issues 317

3.12 Support Vector Domain Description 327

3.12.1 Implementation of Support Vector Machines 333

3.12.2 Architecture of the Ensemble of One-Class Classifiers 334

3.13 Appendix - MATLAB R and other Packages for Pattern Classification 336

3.14 Closure 336

3.14.1 Chapter Summary 336

3.14.2 Further Reading 337

Problems and Exercises 338

References 339

4 Object Detection and Tracking 346

4.1 Introduction 346

4.2 Direct Pixel Classification 346

4.2.1 Ground-Truth Data Collection 347

4.2.2 CASE STUDY - Human Skin Detection 348

4.2.3 CASE STUDY - Pixel Based Road Signs Detection 352

4.2.3.1 Fuzzy Approach 353

4.2.3.2 SVM Based Approach 353

4.2.4 Pixel Based Image